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Recently, it was conjectured that the first generalized Stieltjes constant atrational argument may be always expressed by means of Euler's constant, thefirst Stieltjes constant, the $\Gamma$-function at rational argument(s) andsome relatively simple, perhaps even elementary, function. This conjecture wasbased on the evaluation of $\gamma_1(1/2)$, $\gamma_1(1/3)$, $\gamma_1(2/3)$,$\gamma_1(1/4)$, $\gamma_1(3/4)$, $\gamma_1(1/6)$, $\gamma_1(5/6)$, which couldbe expressed in this way. This article completes this previous study andprovides an elegant theorem which allows to evaluate the first generalizedStieltjes constant at any rational argument. Several related summationformul{\ae} involving the first generalized Stieltjes constant and the Digammafunction are also presented. In passing, an interesting integral representationfor the logarithm of the $\Gamma$-function at rational argument is alsoobtained. Finally, it is shown that similar theorems may be derived for higherStieltjes constants as well; in particular, for the second Stieltjes constantthe theorem is provided in an explicit form.

A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations